The 3PN non-spinning Lagrangian has been studied extensively [1]} and we follow the steps suggested to remove the divergent pieces.
Specifically, following [1]} we add a total derivative term with the complete 3PN Lagrangian
\(\mathcal {L}_{\text{TD}}=\left(\frac{1}{c^6}\right)\frac{1}{\epsilon }\frac{d}{dt}\left[\frac{G_N^3}{r}\Big (c_1 \left(\textbf {v}_{(1)} \cdot \textbf {n}\right) + c_2 \left(\textbf {v}_{(2)} \cdot \textbf {n}\right)\Big )\right]\, ,\) 
